Strong-coupling topological Josephson effect in quantum wires
Flavio S. Nogueira, Ilya Eremin
备注：
20 pages, 6 figures; v3: Extensively revised version; added new figures and corrected the calculation of the current by taking the fixed parity ensemble into account; calculation of the Josephson energy levels remains unchanged
We investigate the Josephson effect for a setup with two lattice quantum wires featuring Majorana zero energy boundary modes at the tunnel junction. In the weak-coupling, the exact solution reproduces the perturbative result for the energy containing a contribution $\sim \pm\cos(ϕ/2)$ relative to the tunneling of paired Majorana fermions. As the tunnel amplitude $g$ grows relative to the hopping amplitude $w$, the gap between the energy levels gradually diminishes until it closes completely at the critical value $g_c=\sqrt{2}w$. At this point the Josephson energies have the principal values $E_{mσ}=2σ\sqrt{2}w\cos[ϕ/6+2π(m-1)/3]$, where $m=-1,0,1$ and $σ=\pm 1$, a result not following from perturbation theory. It represents a transparent regime where three Bogoliubov states merge, leading to additional degeneracies of the topologically nontrivial ground state with odd number of Majorana fermions at the end of each wire. We also obtain the exact tunnel currents for a fixed parity of the eigenstates. The Josephson current shows the characteristic $4π$ periodicity expected for a topological Josephson effect. We discuss the additional features of the current associated with a closure of the energy gap between the energy levels.
We studied anomalous Josephson effect (AJE) in Josephson trijunctions fabricated on Bi$_2$Se$_3$, and found that the AJE in T-shaped trijunctions significantly alters the Majorana phase diagram of the trijunctions, when an in-plane magnetic field is applied parallel to two of the three single junctions. Such a phenomenon in topological insulator-based Josephson trijunction provides unambiguous evidence for the existence of AJE in the system, and might provide an additional knob for controlling the Majorana bound states in the Fu-Kane scheme of topological quantum computation.
Topological Josephson junctions (JJs), which contain Majorana bound states, are expected to exhibit 4$π$-periodic current-phase relation, thereby resulting in doubled Shapiro steps under microwave irradiation. We performed numerical calculations of dynamical properties of topological JJs using a modified resistively and capacitively shunted junction model and extensively investigated the progressive evolution of Shapiro steps as a function of the junction parameters and microwave power and frequency. Our calculation results indicate that the suppression of odd-integer Shapiro steps, i.e., evid…
Quantum spin Hall-superconductor hybrids are promising sources of topological superconductivity and Majorana modes, particularly given recent progress on HgTe and InAs/GaSb. We propose a new method of revealing topological superconductivity in extended quantum spin Hall Josephson junctions supporting `fractional Josephson currents'. Specifically, we show that as one threads magnetic flux between the superconductors, the critical current traces an interference pattern featuring sharp fingerprints of topological superconductivity---even when noise spoils parity conservation.
The promise of fault-tolerant quantum computing has made topological superconductors the focus of intense research during the past decade. In this context, topological Josephson junctions based on nanowires or on topological insulators provide an alternative route for probing topological superconductivity. As a hallmark of their topological nature, such junctions exhibit a ground-state fermion parity that is $4π$-periodic in the superconducting phase difference $ϕ$. Finding unambiguous experimental evidence for this $4π$-periodicity still proves a difficult task, however. Here we propose a top…
The anomalous proximity effect in dirty superconducting junctions is one of most striking phenomena highlighting the profound nature of Majorana bound states and odd-frequency Cooper pairs in topological superconductors. Motivated by the recent experimental realization of planar topological Josephson junctions, we describe the anomalous proximity effect in a superconductor/semiconductor hybrid, where an additional dirty normal-metal segment is extended from a topological Josephson junction. The topological phase transition in the topological Josephson junction is accompanied by a drastic chang…
We study the thermodynamic properties of topological Josephson junctions using a quantum spin Hall (QSH) insulator-based junction as an example. In particular, we propose that phase-dependent measurements of the heat capacity offer an alternative to Josephson-current measurements to demonstrate key topological features. Even in an equilibrium situation, where the fermion parity is not conserved, the heat capacity exhibits a pronounced double peak in its phase dependence as a signature of the protected zero-energy crossing in the Andreev spectrum. This double-peak feature is robust against chan…
Joule overheating poisons the fractional ac Josephson effect in topological Josephson junctions
在三维拓扑绝缘体（TI）表面上设计的拓扑约瑟夫森结在常规Andreev束缚态的连续体中具有马约拉纳束缚态（MBS）。这些MBS的不同特征在于$4π$ -它们的能量-相位关系的周期性，其产生分数交流约瑟夫森效应和在$r\!f$照射下的奇夏皮罗台阶的抑制。然而，最近的实验表明，有几个，或者只是第一个，奇怪的夏皮罗步骤是失踪，对解释投下怀疑。在这里，我们表明，约瑟夫森结上定制的大带隙三维TI Bi $_2$ Se $_3$表现出分数交流约瑟夫森效应作用于第一夏皮罗步骤。用一个改进的反向分流结模型，我们证明了高阶奇Shapiro台阶的弹性可以用焦耳过热驱动的热中毒来解释。此外，我们发现在夏皮罗叶之间的节点，这提供了一个直接的和新颖的签名的MBS所携带的电流的残余超电流。我们的研究结果展示了拓扑约瑟夫森结中的热效应的关键作用，并支持马约拉纳起源的奇数夏皮罗步骤的部分抑制。
Josephson junctions based on three-dimensional topological insulators offer intriguing possibilities to realize unconventional $p$-wave pairing and Majorana modes. Here, we provide a detailed study of the effect of a uniform magnetization in the normal region: We show how the interplay between the spin-momentum locking of the topological insulator and an in-plane magnetization parallel to the direction of phase bias leads to an asymmetry of the Andreev spectrum with respect to transverse momenta. If sufficiently large, this asymmetry induces a transition from a regime of gapless, counterpropag…
We consider a long topological Josephson junction formed on a conducting 2D surface of a 3D topological insulator (TI). The superconducting correlations are proximity-induced by s-wave superconductors covering the surface. The 1D spacing between the coverings is either unfilled or filled by a magnetic insulator. Generally, the Josephson current mediated by the TI surface is determined by scattering modes as well as by the states localized around the junction (Andreev bound states or Andreev bands). We find out that it is crucial to take into account both contributions to determine the current-…
An odd-occupied quantum dot in a Josephson junction can flip transmission phase, creating a π-junction. When the junction couples topological superconductors, no phase flip is expected. We investigate this and related effects in a full-shell hybrid interferometer, using gate voltage to control dot-junction parity and axial magnetic flux to control the transition from trivial to topological superconductivity. Enhanced zero-bias conductance and critical current for odd parity in the topological phase reflects hybridization of the confined spin with zero-energy modes in the leads.
The fractional Josephson effect is known to be a characteristic phenomenon of topological Josephson junctions hosting Majorana zero modes (MZMs), where the Josephson current has a $4π$ (rather than a $2π$) periodicity in the phase difference between the two topological superconductors. We introduce a one-dimensional model of a topological superconductor/normal-metal/superconductor (SNS) junction with the normal-metal (N) region of finite length, which is intermediate regime between the short- and long-junction limits. Assuming weak tunneling at the SN interfaces, we investigate resonance and f…
Wires made of topological insulators (TI) are a promising platform for searching for Majorana bound states. These states can be probed by analyzing the fractional ac Josephson effect in Josephson junctions with the TI wire as a weak link. An axial magnetic field can be used to tune the system from trivial to topologically nontrivial. Here we investigate the oscillations of the supercurrent in such wire Josephson junctions as a function of the axial magnetic field strength and different contact transparencies. Although the current flows on average parallel to the magnetic field we observe $h/2e…
Superconducting circuits incorporating Josephson tunnel junctions are widely used for fundamental research as well as for applications in fields such as quantum information and magnetometry. The quantum coherent nature of Josephson junctions makes them especially suitable for metrology applications. Josephson junctions suffice to form two sides of the quantum metrology triangle, relating frequency to either voltage or current, but not its base, which directly links voltage to current. We propose a five Josephson tunnel junction circuit in which simultaneous pumping of flux and charge results i…
We theoretically investigate how the spin susceptibility of a planar Josephson junction is affected when the system transits into the topological superconducting state. We show that the magnetic flux and magnetic field dependence of the spin susceptibility closely maps the phase diagram of the system. In the absence of an external magnetic flux the system can self-tune into the topological superconducting state by minimizing its free energy. Self-tuned topological transitions are accompanied by sharp peaks in the spin susceptibility, which can therefore be use as measurable fingerprints for de…
